HW 1
Write and submit:
(1) 2, 3, 5 on page 22,
(2) (a) Suppose S is a subset of real numbers R and R, and let
T = cfw_s + | s S. Show that if S has the least upper bound,
sup S, then T also has the least upper bound, sup T . Moreover,
sup T = sup S + .
(
Analysis I: Solutions to PSET 6
Problem 1
(a) Recall from Rudin Theorem 3.3 that the limit of a product is the product
of the limits if the individual limits exist and hence
n
n
lim n n .
lim 2n = lim 2
n
n
n
From Rudin Theorem 3.20, we nd that limn n 2 =
Analysis I: Solutions to PSET 8
Rudin 4.7
The arithmetic/geometric mean inequality shows that f is bounded:
|xy 2 |/2
|xy 2 |
= 2
x2 + y 4
(x + y 4 )/2
|xy 2 |/2
1
= .
2y4
2
x
|f (x, y)| =
However, f is not continuous the limit of f along x = y 2 at (0, 0
Solutions to Assignment 9
1. Rudin 5.2
Let a < x < y < b. Then, f is continuous on [x, y] and dierentiable on
(x, y). Thus, by the Mean Value Theorem, there exists c (x, y) such that
f (y) f (x) = (y x)f (c) > 0, where the inequality follows because f (c)
Analysis I: Solutions to PSET 10
Rudin 6.4
For a < b R let f : [a, b] R be 0 for all irrationals and 1 for all rationals.
The density of Q and R \ Q in R force
n
n
U (P, f ) =
sup
xi1 xxi
i=1
n
L(P, f ) =
inf
xi1 xxi
i=1
xi = b a
f (x) xi =
i=1
n
0 xi = 0
Fall 2014: Intro-Modern Analysis I
Practice Exam I
Name (Last, First)/ UNI:
Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
NO calculators or other electronic de
Fall 2014: Intro-Modern Analysis I
Practice Exam II
Name (Last, First)/ UNI:
Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
NO calculators or other electronic d
Fall 2014: Intro-Modern Analysis I
Practice Final
Name (Last, First):
Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
NO calculators or other electronic devices,
Fall 2014: Intro-Modern Analysis I
Practice Final Name (Last, First):
0 Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
0 NO calculators or other electro
ANALYSIS I
9
9.1
The Cauchy Criterion
Cauchys insight
Our diculty in proving an is this: What is ? Cauchy saw that it was enough to
show that if the terms of the sequence got suciently close to each other. then completeness
will guarantee convergence.
Rem
Fall 2013: Intro-Modern Analysis I
Practice Exam II
Name (Last, First):
Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
NO calculators or other electronic device
Solutions to Assignment 5
1. Rudin 2.15
Dene En := (0, 1/n), n N. Then En is bounded, and for any nite subcollection cfw_Ei iI , let n = miniI cfw_i. Then iI Ei = (0, 1/n) = . Yet we have
that n=1 En = since if 0 < r R we can choose n N such that 1/n < r.
Analysis I: Solutions to PSET 4
Problem 1
Suppose for the sake of contradiction that E F is disconnected, i.e. there
exist opens A, B M such that AB = E F and AB = . Then (AE)
and (B E) provide a separation for E. Since E is connected, we nd, say,
that A
Solutions to Assignment 3
1. Rudin 2.11
d1
d2
d3
d4
d5
is
is
is
is
is
not a metric, since d1 (2, 0) = 4 > 2 = d1 (2, 1) + d1 (1, 0)
a metric.
not a metric, since d3 (1, 1) = 0
not a metric, since d4 (2, 1) = 0
a metric.
2.
Let f, g, h be continuous and re
HW 4
Write and submit:
1. If E and F are connected subsets of M with E F = , show that
E F is connected.
2. 16 on page 44.
3. If K is nonempty compact subset of R, show that sup K and inf K are
elements of K.
4. If A is compact in M and B is compact in N
HW 2
Write and submit:
(1) 17 on page 23,
(2) 2 on page 43,
(3) Show that (0, 1) is equivalent to [0, 1] and also equivalent to R.
(4) Let cfw_En , n = 1, 2, 3, be a sequence of countable sets, and put
S = E1 E2 En . Show that S is uncountable. Prove tha
HW 3
Write and submit:
1. 11 on page 44.
2. Check that d(f, g) = maxaxb |f (x)g(x)| denes a metric on C ([a, b]),
the collection of all continuous and real valued functions dened on the
closed interval [a, b].
3. Prove the following two statements:
1
1
1
HW 6
Write and submit:
1. Prove the following two limits.
n2
n
2n = 1, [b.] lim n = 0.
[a.] lim
n 2
n
2. If cfw_xn is a sequence of real numbers and dene its arithmetic means
an by
x0 + x1 + + xn
an =
, n = 0, 1, 2, 3, .
n+1
(a). If lim xn = a, prove th
HW 5
Write and submit:
1. 15 on page 44.
2. 1 on page 78.
3. 3 on page 78.
4. If xn x in (M, d), show that d(xn , y) d(x, y) for any y M .
More generally, if xn x and yn y, show that d(xn , yn ) d(x, y).
Practice:
1. 26 on page 45,
2. A sequence cfw_xn
HW 7
Write and submit:
1. 9 on page 79.
2. 12 on page 79.
3. 1 on page 98.
4. Let f : R R be continuous.
(a). If f (0) > 0, show that f (x) > 0 for all x in some open interval
(a, a).
(b). If f (x) 0 for every rational x, show that f (x) 0 for all real x
HW 10
Write and submit:
1. 4 on page 138,
2. 7 on page 138,
3. 8 on page 138,
4. 10 (a), (b), (c) on page 139. (In Part (c), the problem says that
f and g are complex functions, you just need to consider the case f
and g are real-valued functions. Indeed
HW 8
Write and submit:
1. 7 on page 99,
2. 8 on page 99,
3. 14 on page 100,
4. 18 on page 100.
Practice:
1. Read Theorem 4.20 on page 91 and try to understand the examples
given in the proof.
2. Read Theorem 4.29 and its corollary on page 96.
3. 10 on p
Solutions to Assignment 1
Part (1)
Exercise 2
Assume by contradiction that there exists a rational number m/n with m, n = 0
without common factors such that m2 /n2 = 12, We have m2 = 12n2 . Hence 3
and 2 divide m2 , thus 6 divides m. Call m = 6m1 and then
Analysis I: Solutions to PSET 2
Rudin 1.17
Take two vectors x, y Rk . We claim that
|x + y|2 + |x y|2 = 2|x|2 + 2|y|2 ,
which follows by the commutativity and distributivity of the dot product,
|x + y|2 + |x y|2 = (x + y) (x + y) + (x y) (x y)
= x x + 2x
Math 230B Homework 2
Erik Lewis April 5, 2007
1
Zhou
3. Prove that if f is integrable on [0,1] and f is continuous at for 0 x 1, then
1
1 2
with f ( 1 ) > 0 and f (x) 0 2
f (x)dx > 0.
0
Proof: From our rst homework Spivak Question 15, if f is continuous,